Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
5778623 | Advances in Mathematics | 2017 | 42 Pages |
Abstract
We study the enumeration of diagonally and antidiagonally symmetric alternating sign matrices (DASASMs) of fixed odd order by introducing a case of the six-vertex model whose configurations are in bijection with such matrices. The model involves a grid graph on a triangle, with bulk and boundary weights which satisfy the Yang-Baxter and reflection equations. We obtain a general expression for the partition function of this model as a sum of two determinantal terms, and show that at a certain point each of these terms reduces to a Schur function. We are then able to prove a conjecture of Robbins from the mid 1980's that the total number of (2n+1)Ã(2n+1) DASASMs is âi=0n(3i)!(n+i)!, and a conjecture of Stroganov from 2008 that the ratio between the numbers of (2n+1)Ã(2n+1) DASASMs with central entry â1 and 1 is n/(n+1). Among the several product formulae for the enumeration of symmetric alternating sign matrices which were conjectured in the 1980's, that for odd-order DASASMs is the last to have been proved.
Related Topics
Physical Sciences and Engineering
Mathematics
Mathematics (General)
Authors
Roger E. Behrend, Ilse Fischer, Matjaž Konvalinka,